Session overview

We often collect data from two or more groups. Group allocations can be stored as categorical variables, and we often want to explore and compare differences between groups, using these variables.

In this session we cover

  • how R stores different types of data,
  • look at how this affects the way we plot and summarise data
  • introduce the Bayesian t-test to quantify the evidence for differences between groups

Different types of variables and visual scales

The video introduces the link types of variable (e.g. continuous, categorical, text), the data-types which R uses to store them, and the way that ggplot presents them on the scales of a plot.

  • Common types of variable are: numeric, categorical and text (string)
  • Internally, R stores data in a number of different data types.
  • These data types mostly match up the different types of variable — but not always, so watch out!
  • For example, sometimes numeric data can get stored as text by accident (we would need to convert this)
  • Categorical variables can be stored as either factors or as text/strings (again, we can convert between them as needed)
  • ggplot (and other R functions) use data-types as a clue to choose defaults for the scales of your graphs
  • Normally the defaults are good; sometimes it’s helpful to manually adjust the scale by switching the data type

The following R code is used in the video:

# load the tidyverse package
library(tidyverse)

# ... etc etc

Data types

Data comes in all shapes and sizes, but and important distinction researchers make is between types of variable.

You might have seen terms like these:

  • interval or continuous variables data (also called real numbers)
  • ordinal variables (e.g. Likert style 1-7 responses, sometimes called factors in experimental designs)
  • count variables (whole numbers greater than or equal to zero)
  • nominal or categorical variables, which are also sometimes

These data-types mostly match up the different types of variable — and the names will be the same — but it’s not always the case. Sometimes we need to convert between data types.

In R there are three main data-types you need to know about:

  • Numeric data, which are stored as a ‘double’, abbreviated dbl. ‘Double’ means ‘double precision number’, which is computer speak for ‘any kind of real number, even a very large one’.

  • Categorical data, which is stored as a factor

  • Text, which is stored with the ‘character’ data-type (abbreviated chr)

You might also encounter these data types, but we don’t specifically need them for this course:

  • Boolean data (true/false values)
  • Dates (a special kind of numeric data, which R formats nicely as a date for us)
  • Ordinal variables (a special type of factor where the categories are ordered)

If you have a variable in R you can use the typeof function to check which data-type it is. For example:

typeof(1)
[1] "double"
typeof("apple")
[1] "character"

You can also see which data-type is used to store a variable when using the glimpse command you saw in session 1 (e.g. here).

iris %>% glimpse
Rows: 150
Columns: 5
$ Sepal.Length <dbl> 5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9, 5.4, 4…
$ Sepal.Width  <dbl> 3.5, 3.0, 3.2, 3.1, 3.6, 3.9, 3.4, 3.4, 2.9, 3.1, 3.7, 3…
$ Petal.Length <dbl> 1.4, 1.4, 1.3, 1.5, 1.4, 1.7, 1.4, 1.5, 1.4, 1.5, 1.5, 1…
$ Petal.Width  <dbl> 0.2, 0.2, 0.2, 0.2, 0.2, 0.4, 0.3, 0.2, 0.2, 0.1, 0.2, 0…
$ Species      <fct> setosa, setosa, setosa, setosa, setosa, setosa, setosa, …

In the glimpse output you can see the variable names listed on the left, followed by grey text surrounded by angle brackes, e.g.: <dbl> which is the abbreviated data-type.

In this built-in dataset, most of the data is numeric (dbl), but the Species variable is categorical, and stored as a factor (fct).

Data types and scales on graphs

If we look at the mtcars data we can see that all the variables are stored as numeric data (dbl):

mtcars %>% glimpse()
Rows: 32
Columns: 11
$ mpg  <dbl> 21.0, 21.0, 22.8, 21.4, 18.7, 18.1, 14.3, 24.4, 22.8, 19.2, 17.8…
$ cyl  <dbl> 6, 6, 4, 6, 8, 6, 8, 4, 4, 6, 6, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 8…
$ disp <dbl> 160.0, 160.0, 108.0, 258.0, 360.0, 225.0, 360.0, 146.7, 140.8, 1…
$ hp   <dbl> 110, 110, 93, 110, 175, 105, 245, 62, 95, 123, 123, 180, 180, 18…
$ drat <dbl> 3.90, 3.90, 3.85, 3.08, 3.15, 2.76, 3.21, 3.69, 3.92, 3.92, 3.92…
$ wt   <dbl> 2.620, 2.875, 2.320, 3.215, 3.440, 3.460, 3.570, 3.190, 3.150, 3…
$ qsec <dbl> 16.46, 17.02, 18.61, 19.44, 17.02, 20.22, 15.84, 20.00, 22.90, 1…
$ vs   <dbl> 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0…
$ am   <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0…
$ gear <dbl> 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3…
$ carb <dbl> 4, 4, 1, 1, 2, 1, 4, 2, 2, 4, 4, 3, 3, 3, 4, 4, 4, 1, 2, 1, 1, 2…

This is fine if we want to make a scatter plot (here of ‘miles per gallon’ vs weight of the car):

mtcars %>%
  ggplot(aes(wt, mpg)) +
    geom_point()

In this plot both the x and y axes are continuous. That is, they are numeric variables, using real numbers.

However, if we want to make a boxplot of the mpg variable using am as the x axis then we have a problem:

mtcars %>%
  ggplot(aes(am, mpg)) +
  geom_boxplot()
Warning: Continuous x aesthetic -- did you forget aes(group=...)?

We might have expected to see;

  • miles per gallon on the y axis
  • two separate boxes, one for automatic cars and another for manual.

This doesn’t work as expected though.

Because ggplot has seen that am is stored as numeric data, it creates a continuous scale on the x axis, and draws a single box at the midpoint of all the values of am. Because am ranges from 0 to 1, this box appears at 0.5.

As glimpse showed us, the variable am is stored as numeric data, with type dbl (short . However, we really want to use am as a categorical variable. So we should store it as a factor. If we convert it to a factor then our plot will work properly.

We can use the command factor(am) to tell R that the x-axis is a factor:

mtcars %>%
  ggplot(aes(factor(am), mpg)) +
  geom_boxplot()

This gives us the boxplot we were expecting. The only change here is to replace am with factor(am). This tells R to convert the variable am to a factor. ggplot can then draw the x axis correctly.

You need to learn these datatypes and abbreviations:

| Data type | Abbreviation | Used for |
| double | dbl | Numeric data (e.g. interval or continuous variables) | | character| chr | text data, and sometimes also categorical variables | |||| |||| |||| ||||
  • Hide this page and test a friend or someone next to you in the room on what each of them means.
  • Repeat this in 20 minutes time to check you still have it (spaced repetition is effective).

Exercise XXX

Use mtcars to make a boxplot showing miles per gallon on the y axis, and number of gears the car has on the x axis (gear).

Your plot should look like this:

XXX ADD EXTENSION EXERCISES TO DO THE SAME WITH COLOR SCALES>>> E.G.

mtcars %>% 
  ggplot(aes(wt, mpg, color=gear)) + 
  geom_point()


mtcars %>% 
  ggplot(aes(wt, mpg, color=factor(gear))) + 
  geom_point()

Exploring grouped data

The video explains and gives examples showing that:

  • Datasets often contain categorical variables
  • We often want to compare statistics (like averages) between categories
  • The group_by function is a quick way to combine filtering and summarising
  • group_by creates a grouped dataframe
  • Using grouped dataframes with other functions (e.g. summarise) applies them once-per-group
  • The result is always a new dataframe

The following R code is used in the video:

# mtcars has the `gear`, `cyl` and `am` variables, which could be treated as 
# either categorical or numeric
mtcars %>% select(gear, cyl, am) %>% head
                  gear cyl am
Mazda RX4            4   6  1
Mazda RX4 Wag        4   6  1
Datsun 710           4   4  1
Hornet 4 Drive       3   6  0
Hornet Sportabout    3   8  0
Valiant              3   6  0


# we previously made a box-plot broken down by a category
mtcars %>%
  ggplot(aes(factor(gear), mpg)) + 
  geom_boxplot()


# we can use filter to calculate averages for each category
mtcars %>%
  filter(gear == 4 ) %>% 
  summarise(mean(mpg))
  mean(mpg)
1  24.53333

mtcars %>%
  filter(gear == 5 ) %>% 
  summarise(mean(mpg))
  mean(mpg)
1     21.38

# ... and so on. However, this gets repetitive with many groups.


# Instead we can use group_by to make a table with a row for each group
mtcars %>% 
  group_by(gear) %>% 
  summarise(mean(mpg))
# A tibble: 3 x 2
   gear `mean(mpg)`
  <dbl>       <dbl>
1     3        16.1
2     4        24.5
3     5        21.4


# We can add standard deviations (or other stats) to the same table and give each column a name
mtcars %>% 
  group_by(gear) %>% 
  summarise(Mean = mean(mpg), SD = sd(mpg))
# A tibble: 3 x 3
   gear  Mean    SD
  <dbl> <dbl> <dbl>
1     3  16.1  3.37
2     4  24.5  5.28
3     5  21.4  6.66

Complete text of the video here. Markdown can be used. Keep blank lines above and below

Use the built-in iris dataset

Use group_by to calculate the average Sepal.Length of each Species of flower.

Comparing the averages of groups

BSc students will have encountered some of this last year; but there’s new stuff too, and important revision.

The video explains that:

  • t-tests allow us to compare the average score for two groups
  • A Bayes Factor (from a Bayesian t-test) compares two different hypotheses: H1 (there groups are different) vs. H0 (the groups are identical)
  • Use the ttestBF function (remember to load the BayesFactor package first)
  • Large Bayes Factors (e.g. > 10) mean we have a lot of evidence there is a difference between groups
  • Very small Bayes Factors (e.g. < .1) mean we have strong evidence there is NO difference
  • Bayes factors \(>3\) or < \(\frac{1}{3}\) provide limited evidence
  • Bayes factors between \(\frac{1}{3}\) and \(3\) are inconclusive

The following R code is used in the video:

# how much more fuel efficient are manual cars?
mtcars %>% 
  group_by(am) %>% 
  summarise(mean(mpg))
# A tibble: 2 x 2
     am `mean(mpg)`
  <dbl>       <dbl>
1     0        17.1
2     1        24.4

# about 7 mpg


# we can use a boxplot to see how much overlap there is between the groups
mtcars %>% 
  ggplot(aes(factor(am), mpg)) +
  geom_boxplot()


# it looks like they don't overlap too much, which means there probably is a 
# real difference but we want to use a Bayesian t-test and a "Bayes Factor" to 
# quantify how much more likely it is that: 
# A) there really is a difference vs.
# B) there's really no difference and this is chance variation

library(BayesFactor)
ttestBF(formula = mpg ~ am, data=mtcars)
Bayes factor analysis
--------------
[1] Alt., r=0.707 : 86.58973 ±0%

Against denominator:
  Null, mu1-mu2 = 0 
---
Bayes factor type: BFindepSample, JZS


# the number 86.58973 is our Bayes Factor. This is > 10 so we have strong 
# evidence for a difference (Hypothesis A)

XXX FLESH THIS OUT TO BE MORE CONVERSATIONAL AND INTEGRATE/EXPLAIN THE CODE BELOW

  • A ‘Bayesian t-test’ is a procedure which tells us how likely it is that two groups have a different average for a continuous variable.

  • A Bayesian t-test imagines two different worlds: In world A) the groups truly do have a different average. In world B) the groups are really identical, with just chance errors leading to small differences between them in observations we make.

  • We can get an intuition for how much evidence there is for a difference by looking at how much the distributions of the two groups overlap [e.g. a boxplot].

  • After collecting data we use a Bayes Factor to quantify how much more likely world A is v.s. world B (or the reverse).

  • That is: given the data we have collected and the differences (big or small) that we see, we can estimate “how much more likely is it that there really is a difference between the groups, v.s. that there is no difference”

  • Another way to think of these alternative worlds (A and B) is as different hypotheses about how our data came about.

  • Sometimes we talk about comparing an experimental hypothesis (H1) with a null hypothesis (H0)

  • When we compare two groups (with a t-test) we have a very simple experimental hypothesis — simply, that the groups are not identical

  • If we run a t test and find a large Bayes Factor, this is evidence for that experimental hypothesis, as compared with the null hypothesis (that the groups are the same)

Check your knowledge

Write an answer to each of these questions in the Check your knowledge section of your workbook. The answers are revealed in Session 4.

  • What is the difference between a dbl and a fct or ord?
  • Give an example of when the difference between dbl and fct matters when making a plot? (include code examples for this if you can)
  • How can you convert a variable from a dbl to a fct?